Russell's paradox and Godel's incompleteness theorem prove that the CTMU is invalid.
In order to be consistent, mathematics must possess a kind of algebraic closure, and to this extent must be globally self-referential. Concisely, closure equals self-containment with respect to a relation or predicate, and this equates to self-reference. E.g., the self-consistency of a system ultimately equates to the closure of that system with respect to consistency, and this describes a scenario in which every part of the system refers consistently to other parts of the system (and only thereto). At every internal point (mathematical datum) of the system mathematics, the following circularity applies: "mathematics refers consistently to mathematics". So mathematics is distributively self-referential, and if this makes it globally vulnerable to some kind of implacable "meta-mathematical" paradox, all we can do in response is learn to live with the danger. Fortunately, it turns out that we can reason our way out of such doubts...but only by admitting that self-reference is the name of the game. Source *'To demonstrate the existence of undecidability, Gödel used a simple trick called self-reference. Consider the statement “this sentence is false.” It is easy to dress this statement up as a logical formula. Aside from being true or false, what else could such a formula say about itself? Could it pronounce itself, say, unprovable? Let’s try it: "This formula is unprovable". If the given formula is in fact unprovable, then it is true and therefore a theorem. Unfortunately, the axiomatic method cannot recognize it as such without a proof. On the other hand, suppose it is provable. Then it is self-apparently false (because its provability belies what it says of itself) and yet true (because provable without respect to content)! It seems that we still have the makings of a paradox…a statement that is "unprovably provable" and therefore absurd. ' *What if we now introduce a distinction between levels of proof, calling one level the basic or "language" level and the other (higher) level the "metalanguage" level? Then we would have either a statement that can bemetalinguistically proven to be linguistically unprovable, and thus recognizable as a theorem conveying valuable information about the limitations of the basic language, or a statement that cannot be metalinguistically proven to be linguistically unprovable, which, though uninformative, is at least not a paradox. Presto: self-reference without the possibility of paradox! *Such paradoxes are properly viewed not as static objects, but as dynamic alternations associated with a metalinguistic stratification that is constructively open-ended but transfinitely closed (note that this is also how we view the universe). Otherwise, the paradox corrupts the informational boundary between true and false and thus between all logical predicates and their negations, which of course destroys all possibility of not only its cognitive resolution, but cognition and perception themselves. Yet cognition and perception exist, implying that nature contrives to resolve such paradoxes wherever they might occur. In fact, the value of such a paradox is that it demonstrates the fundamental necessity for reality to incorporate a ubiquitous relativization mechanism for its resolution, namely the aforementioned metalinguistic stratification of levels of reference (including levels of self-reference, i.e. cognition). A paradox whose definition seems to preclude such stratification is merely a self-annihilating construct that violates the "syntax" (structural and inferential rules) of reality and therefore lacks a real model. *In other words, to describe reality as cognitive, we must stratify cognition and organize the resulting levels of self-reference in a self-contained mathematical structure called Self-Configuring Self-Processing Language or SCSPL. SCSPL, which consists of a monic, recursive melding of information and cognition called infocognition, incorporates a metalogical axiom,Multiplex Unity or MU, that characterizes the universe as a syndiffeonic relation or "self-resolving paradox" (the paradox is "self-resolving" by virtue of SCSPL stratification). A syndiffeonic relation is just a universal quantum of inductive and deductive processing, i.e. cognition, whereby "different" objects are acknowledged to be "the same" with respect to their mutual relatedness. Source *One can’t even conceive of logic without applying a distributed "power-set template" to its symbols and expressions, and such templates clearly perform a syntactic function.